The interpolation based algebraic decoding for Reed-Solomon (RS) codes can correct errors beyond half of the code's minimum Hamming distance. Using soft information, the algebraic soft decoding (ASD) further improves the decoding performance. This paper presents a unified study of two classical ASD algorithms in which the computationally expensive interpolation is solved by the module minimization (MM) technique. An explicit module basis construction for the two ASD algorithms will be introduced. Compared with Koetter's interpolation, the MM interpolation enables the algebraic Chase decoding and the Koetter-Vardy decoding perform less finite field arithmetic operations. Re-encoding transform is applied to further reduce the decoding complexity. Computational cost of the two ASD algorithms as well as their re-encoding transformed variants are analyzed. This research shows re-encoding transform attributes to a lower decoding complexity by reducing the degree of module generators. Furthermore, Monte-Carlo simulation of the two ASD algorithms has been performed to show their decoding and complexity competency.